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Find the value of \(x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}\)

Hint(s):What's the relationship between \(x\) and \(x+1\)?

Guest Dec 21, 2018

#1**0 **

Here is a computer summation of your continued fraction:

=1E100; listforeach(b,reverse(1,2,2,2.........etc ), c=b + 1/c);printc

c=1.4142135623730950488016887242097

The result is sqrt(2)

Guest Dec 21, 2018

edited by
Guest
Dec 21, 2018

edited by Guest Dec 21, 2018

edited by Guest Dec 21, 2018

#2**+2 **

\(x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}\\ x = -1+2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}\\ x +1= 2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}\\ x +1= 2 + \cfrac{1}{x+1}\\ \)

\(x +1= 2 + \cfrac{1}{x+1}\\ x-1= \cfrac{1}{x+1}\\ x^2-1=1\\ x^2=2\\ x=\pm\sqrt2 \)

I do not know why the answer cannot be -sqrt2 but if memory serves me correctly the only correct answer is +sqrt2

I'll continure thinking about that one.

Melody Dec 21, 2018